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Decreasing Chain Sequence
For tree-based definitions, go to Decreasing Chain Sequence/Tree-Based Definitions. For an analysis, go to Decreasing Chain Sequence/Analysis. I will be using XNS from now on, however DCS was the inspiration for some of my other notations such as SKCSN and XNS (all 3 versions of XNS) -C7X DCS0 Definition Definition of chains: \(0\in\text{Chain}\) For \(A_1,A_2\in\text{Chain}\) where \(A_1,A_2\neq 0\), \(A_1A_2\in\text{Chain}\) For \(A_1\in\text{Chain}\), \(A_1\in\text{Chain}\) Define a relation \(<\) over \(\text{Chain}\) such that: #\(a_1a_20\) as the longest possible length of a (A,n)-based chain sequence. Explanation This is a weaker version of DCS1. This should reach \(\varepsilon_0\)? DCS1 DCS1.0 Definition Definition of chains: \(0\in\text{Chain}\) For \(A\in\text{Chain}\), \(A\in\text{Chain}\) For \(A_1,A_2\in\text{Chain}\) where \(A_1,A_2\neq 0\), \(A_1A_2\in\text{Chain}\) For \(A_1,A_2\in\text{Chain}\), \(A_1,A_2\in\text{Chain}\) Define a relation \(<\) over \(\text{Chain}\) such that: #\(a_1a_20\) as the longest possible length of a (A,n)-based chain sequence. Explanation A chain is a sequence of other chains that are enclosed in brackets or concatenated together. The base case is that 0 is a chain. Brackets with 3 or more elements aren't allowed, and 0's can't be concatenated. Examples of chains: "0", "0", "00", "00", "0,0", "[0,00]", "[0,0,0]", "0,0[0,0,0]0,0", "0,[[[0,0]],0]" Examples of non-chains: "00" (0 concatenation), "[0,0,0]" (more than 3 chains in brackets) Comparing chains: # There are 2 cases where AB Definition Definition of chains: \(0\in\text{Chain}\) For \(A_1,A_2\in\text{Chain}\) where \(A_1,A_2\neq 0\), \(A_1A_2\in\text{Chain}\) For \(A_1,A_2\in\text{Chain}\), \(A_1,A_2\in\text{Chain}\) Define a relation \(<\) over \(\text{Chain}\) such that: #\(a_1a_20\) as the longest possible length of a (A,n)-based chain sequence. Explanation A chain is a sequence of other chains that are enclosed in brackets or concatenated together. The base case is that 0 is a chain. Each set of brackets contains exactly 2 elements, and 0's can't be concatenated. Examples of chains: "0", "0,0", "0,00", "[0,0,0]", "0,[0,0,0][0,0,0]0" Examples of incorrect chains: "0" (doesn't have 2 elements), "0,0,00,0" (doesn't have 2 elements), "00" (no concatenation of 0s), "00" (doesn't have 2 elements, also no concatenation of 0s) Comparing chains: # There are 2 cases where AB1\), \(C(y,h,k)\) has a zero count of at most \((y+k)!\) A fast-growing function \(DCS(A,n)\) can be defined for a chain \(A\) and a natural number \(n>0\) as the longest possible length of a (A,n)-based chain sequence. Example (Currently incorrect) For example, show that \([0,[0,0,[0,0,0]]]<[0,[0,0,[0,[0,0,[0,[0,0,0]]]]]]\) is true. Some chains/parts of chains are color-coded for clarity. [0,[0,0,[0,0,0]]]<[0,[0,0,[0,[0,0,[0,[0,0,0]]]]]] Applying rule 1 yields the following: [0,[0,0,[0,0,0]]]<[0,[0,0,[0,[0,0,[0,[0,0,0]]]]]] iff 0<0 or (0=0 and [0,0,[0,0,0]]<[0,0,[0,[0,0,[0,[0,0,0]]]]]) --> Because the statement on the left side of the "or" is false, and the statement on the left side of the "and" is true, the only way that the above inequality can be true is if the following is also true: [0,0,[0,0,0]]<[0,0,[0,[0,0,[0,[0,0,0]]]]] Applying rule 1 yields the following: [0,0,[0,0,0]]<[0,0,[0,[0,0,[0,[0,0,0]]]]] iff 0,0<0,0 or (0,0=0,0 and [0,0,0]<[0,[0,0,[0,[0,0,0]]]]) --> Because the statement on the left side of the "or" is false, and the statement on the left side of the "and" is true, the only way that the above inequality can be true is if the following is also true: [0,0,0]<[0,[0,0,[0,[0,0,0]]]] Applying rule 1 yields the following: [0,0,0]<[0,[0,0,[0,[0,0,0]]]] iff 0<0 or (0=0 and 0,0<[0,0,[0,[0,0,0]]]) --> Because the statement on the left side of the "or" is false, and the statement on the left side of the "and" is true, the only way that the above inequality can be true is if the following is also true: 0,0<[0,0,[0,[0,0,0]]] Applying rule 1 yields the following: 0,0<[0,0,[0,[0,0,0]]] iff 0<0 or (0=0 and 0<0,0) --> Because the statement on the left side of the "or" is false, and the statement on the left side of the "and" is true, the only way that the above inequality can be true is if the following is also true: 0<0,0 --> The above inequality is true because of rule 3. This also implies that all of the above statements are true, and therefore the original inequality [0,[0,0,[0,0,0]]]<[0,[0,0,[0,[0,0,[0,[0,0,0]]]]]] is true as well. Analysis Analysis of DCS1.2 \(\text{DCS}(0,n)=1 \\ \text{DCS}(0,0,n)=2 \\ \text{DCS}([0,0,0],n)=3 \\ \text{DCS}([0,[ 0,0,0]],n)\approx (n/3)! \\ \text{DCS}([0,[0,[ 0,0,0]]],n)\approx (n/3)!+1 \\ \text{DCS}([0,[0,[0,[ 0,0,0]]]],n)\approx (n/3)!+2 \\ \text{DCS}([0,[0,[0,[0,[ 0,0,0]]]]],n)\approx (n/3)!+3 \\ \text{DCS}([0,[ 0,0,[0,[ 0,0,0]]]],n)\approx (n/3)!\times2 \\ \text{DCS}([0,[0,[ 0,0,[0,[ 0,0,0]]]]],n)\approx (n/3)!\times2+1 \\ \text{DCS}([0,[ 0,0,[0,[ 0,0,[0,[ 0,0,0]]]]]],n)\approx (n/3)!\times3 \\ \text{DCS}([0,[ 0,0,[ 0,0,0]]],n)\approx (n/3)!^2 \\ \) Code function lss(a,b){ //a Define a relation \(<\) over \(\text{Chain}\) such that: #\(a_1a_2